Let z₁, z₂, and z₃ be three complex numbers on the circle |z| = 1 with arg (z₁) = (π)/4, arg (z₂) = 0, and arg (z₃) = (π)/4. If |z₁ z₂ + z₂ z₃ + z₃ z₁|² = α + β √(2), α, β Z, then the value of α² + β² is:

Complex Numbers and Quadratic Equations: JEE Main 2025 Mathematics question

JEE Main 2025 · January, Unspecified (session-level dataset) · Q220

JEE Main 2025January, Unspecified (session-level dataset)MathematicsMediumComplex Numbers and Quadratic Equations264× since 2002

Let

z_1, z_2

, and

z_3

be three complex numbers on the circle

|z| = 1

with

\arg(z_1) = \frac{\pi}{4}

\arg(z_2) = 0

, and

\arg(z_3) = \frac{\pi}{4}

. If

|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{2}

\alpha, \beta \in \mathbb{Z}

, then the value of

\alpha^2 + \beta^2

is:

JEE Main 2025 · January, Unspecified (session-level dataset) · Q220

JEE Main 2025January, Unspecified (session-level dataset)MathematicsMediumComplex Numbers and Quadratic Equations264× since 2002

Let

z_1, z_2

, and

z_3

be three complex numbers on the circle

|z| = 1

with

\arg(z_1) = \frac{\pi}{4}

\arg(z_2) = 0

, and

\arg(z_3) = \frac{\pi}{4}

. If

|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{2}

\alpha, \beta \in \mathbb{Z}

, then the value of

\alpha^2 + \beta^2

is: